Hamiltonian systems, symplectic flows, classical integrable systems

**6**

votes

**2**answers

430 views

### Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems).
A ...

**9**

votes

**2**answers

292 views

### Is every symplectic manifold a Hamiltonian reduction of a cotangent bundle?

Today I heard the claim that in practice, all symplectic manifolds that people care about arise as the Hamiltonian reduction of a cotangent bundle $T^{\ast}(M)$ under the action of a Lie group $G$ ...

**0**

votes

**1**answer

88 views

### Constant symplectic structure

Let $E$ be a Frechet space and $\mathcal{F}$ be a non-degenerate bounded skew symmetric bilinear map $\mathcal{F}: E\times E\to \mathbb R$ on $E$. We can identify $TE$ with $E\times E$, with this ...

**4**

votes

**2**answers

228 views

### Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?

It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...

**1**

vote

**0**answers

126 views

### Lagrangian complement in a symplectic vector bundle

A standard, folk result in symplectic geometry states that:
in a symplectic vector bundle $(E,\pi,B,\omega)$, any lagrangian subbundle $L$ admits a lagrangian complement $L'$.
Having to use this ...

**4**

votes

**2**answers

324 views

### symplectic structure of tangent bundle of $\mathbb{S}^{n-1}$

It is well known that $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where
$f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$.
Two questions:
1) Is $M$ a ...

**3**

votes

**1**answer

160 views

### Special Lagrangians and fat

I am unable to find the MO comments about the first use of the phrase "fat slags" in an article. On page 26 of this we find "these correspond to thickenings of the
corresponding special Lagrangian ...

**4**

votes

**1**answer

357 views

### Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization.
I don't undrestand the philosophy of following assertion
If $(V,\omega)$ be a symplectic vector space then the quantizations of
$V$ ...

**5**

votes

**1**answer

276 views

### Is there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?

Context
According to Arnol'd, a contact structure on a smooth manifold $M$ is given by a corank 1 tangent distribution $C$ which is maximally non-integrable; this means that, for any local $1$-form ...

**3**

votes

**1**answer

186 views

### What is twisted in a twisted Poisson structure?

The usual definition of a twisted Poisson algebra involving a 3-form
does not seem to refer to a Poisson algebra that is then twisted,
i.e. twisting either the commutative product or the Lie bracket.
...

**28**

votes

**5**answers

870 views

### are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...

**5**

votes

**1**answer

256 views

### Is the derived Fukaya category a derived category in the classical sense?

I am trying to read Seidel's book, and I am confused about the "derived" Fukaya category. If I understand properly, one starts from the $A_{\infty}$-category $\mathfrak{F}(M)$, enlarges it to the ...

**3**

votes

**0**answers

204 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**0**

votes

**1**answer

276 views

### Symplectic structure on $Sym^kG^{\mathbb{C}} $

Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification.
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here ...

**2**

votes

**1**answer

339 views

### looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e,
($J^1M=T^*M×\mathbb{R}$)
Is there something like this identity for higher jet bundle $J^kM$?
I editted ...

**1**

vote

**1**answer

331 views

### pre-quantization of Jet bundle

We know that the notion of Jet bundle $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?

**2**

votes

**1**answer

335 views

### a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?

**1**

vote

**1**answer

94 views

### Orthogonal symplectic classes with respect to intersection product

Let $M$ be a simply connected smooth compact four manifold. Now I am trying to find an example such that
(1) there is two symplectic forms $\omega$ and $\sigma$ on $M$, and
(2) the intersection ...

**2**

votes

**1**answer

200 views

### Complement of Donaldson's symplectic submanifold

I am just starting to learn more symplectic geometry on Stein manifolds. I understand that an important class of Stein manifolds arises as the complement of a Donaldson's codimension two symplectic ...

**4**

votes

**1**answer

222 views

### The Maslov triple product is alternating in its entries

Let $(V,\omega)$ be a $2g$-dimensional symplectic vector space. I'm trying to understand the Maslov triple product. I know that it can be defined in a variety of ways, but for the applications I'm ...

**6**

votes

**1**answer

281 views

### What's the relation between the heat kernel proof of the index theorem and deformation quantization?

In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat ...

**18**

votes

**2**answers

852 views

### The symplectic geometry of cold coconuts

Consider the open set $M \subset \mathbb{C}^{2}$ given by the union of the unit ball $|z_1|^2 + |z_{2}|^2 < 1$ (the coconut) and the cylinder $|z_1| < \epsilon$, $0 < \epsilon < \! \!< ...

**9**

votes

**0**answers

361 views

### Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...

**9**

votes

**1**answer

462 views

### The universal property of the Liouville $1$-form

I am not totally sure if this question is appropriate for MathOverflow, or if it more adeguate to MathStackexchange.
As usual any feedback is welcome.
Introduction
Given an arbitrary smooth manifold ...

**1**

vote

**0**answers

154 views

### Hyperkaehler Structures on the cotangent bundle

Let $M$ be a symplectic manifold (not Kaehler). Does there exists in a neighbourhood of the zero section in the cotangent bundle $T^{*}M$ a Hyperkaehler structure? I know that by the paper by Feix on ...

**0**

votes

**2**answers

287 views

### even dimensional manifold homotopic to a symplectic or complex manifold

I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a ...

**2**

votes

**0**answers

135 views

### Geometric meaning of a certain form in almost-Kähler geometry

I have difficulties finding an appropriate reference for the following question:
Let $(M^{2n},g,J,\omega)$ be a compact almost Kähler manifold. Let $\operatorname{ric}$ the usual Ricci tensor of ...

**0**

votes

**1**answer

186 views

### Coadjoint orbits and homogeneous symplectic $G$-manifolds

We know this important fact from A.A.Kirillov that :
Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...

**1**

vote

**1**answer

99 views

### Monotonicity and perturbation of $J$-holomorphic curves

In a symplectic manifold $(M, \omega)$, $J$-holomorphic curves are special minimal surfaces if $J$ is compatible with the symplectic structure and the metric is induced by $\omega$ and $J$.
Minimal ...

**2**

votes

**1**answer

520 views

### Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $

My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...

**0**

votes

**1**answer

396 views

### fiber bundle on an orbit of $\mathfrak{g}\oplus\mathfrak{g^*}$

Let $G$, be a Lie Group and $\mathfrak{g}$ be its Lie algebra ,i.e, $Lie(G)=\mathfrak{g}$. Let $\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^*}$. Here $X\in \mathfrak{g} $ and $F\in ...

**3**

votes

**2**answers

457 views

### Legendrian Tubular Neighborhood Theorem

Given a Lagrangian submanifold $L\subset(M,\omega)$ of a symplectic manifold, we have Alan Weinstein's celebrated Lagrangian tubular neighborhood theorem. I now look for the analog on Legendrian ...

**6**

votes

**2**answers

300 views

### How many Lagrangian submanifolds?

An $n$-dimensional submanifold $L$ of a symplectic manifold $(M^{2n}, \omega)$ is called Lagrangian if $\omega|_L = 0$. I want to get some feeling about how many Lagrangian submanifolds are.
For each ...

**0**

votes

**0**answers

122 views

### When a hyperplane of symmetric forms is determined by a quadric hypersurface?

Let $L$ be a 2D real vector space, $L^*$ its dual, and $\{V,\omega\}$ the symplectic space with $V=L\oplus L^*$ and $\omega$ unambiguously defined by $\omega(l,\lambda):=\lambda(l)$, for all $l\in L$ ...

**3**

votes

**1**answer

295 views

### Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...

**1**

vote

**1**answer

262 views

### The space of generalized complex structures in sense of N.Hitchin is contractible?

Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...

**2**

votes

**1**answer

112 views

### model co-isotropic submanifold

Just as the zero section in $T^*\mathbb{R}^N$ (equipped with the standard symplectic form) is the "model" / "quintessential" Lagrangian submanifold, does $T^*\mathbb{R}^N$ have a "model" co-isotropic ...

**5**

votes

**2**answers

331 views

### Information from Moment Polytopes

Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^*$. If ...

**8**

votes

**1**answer

349 views

### Does there always exist a line bundle whose Chern class represents an integer symplectic form?

Let $(M, \omega, J)$ be a compact symplectic manifold with a
compatible almost complex structure $J$, such that the symplectic
form determines an integer cohomology class, ie
$$ [\omega] \in H^2(M, ...

**7**

votes

**3**answers

429 views

### Spaces of symplectic embeddings: Bundle? Smoothness?

Let $(M, \omega)$ and $(N, \sigma)$ be two symplectic manifolds, $M$ compact and without boundary. Consider the space $$ \mathcal{E} = \mathrm{Emb}((M, \omega), (N, \sigma)) $$ of all smooth ...

**3**

votes

**1**answer

128 views

### Level sets of Hamiltonians of S^1 actions

Suppose that $(M,\omega)$ is a (connected compact) symplectic manifold with a Hamiltonian $S^1$-action given by Hamiltonian $H$. I would like to find a reference for the fact that every level set of ...

**5**

votes

**4**answers

334 views

### Prequantization and Hilbert space

In the definition of pre-quantization on a symplectic manifold $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ ...

**1**

vote

**3**answers

399 views

### prequantization on $TM \bigoplus T^*M$

Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin ...

**5**

votes

**1**answer

250 views

### Riemannian and symplectic structures

Let $(\mathcal M,g)$ be a smooth Riemannian manifold and $\Delta$ be the standard (positive) Laplace operator given in coordinates by the usual
$$
\Delta=-\vert g\vert^{-1/2}\partial_j(\vert ...

**1**

vote

**1**answer

198 views

### choices of connection in prequantization

In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...

**1**

vote

**1**answer

137 views

### pre-symplectic and foliation and its trajectories

Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?

**1**

vote

**0**answers

239 views

### contact structure on 3 manifolds

every orientable closed 3-manifold admits a contact structure. how we can construct this contact structure? if the manifold is prime what will happen?

**5**

votes

**1**answer

202 views

### Looking for a special rank 2 vector bundle

Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$.
By Riemann-Roch theorem,
$$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$
Question: For which $g$, there is such ...

**3**

votes

**1**answer

157 views

### Can symplectic blow up increase symplectic capacities?

Let $N$ be a symplectic submanifold of $M$. Symplectic blow up of $M$ along $N$ is an operation replacing a tubular neighborhood of $N$ with the projectivization of that neighborhood. So it decreases ...

**1**

vote

**1**answer

378 views

### Every real-holomorphic Hamiltonian vector field on a Kähler manifold is Killing (and preserves curvature), yes?

Following the notation of Moroianu's Lectures on Kähler Geometry, we let $(M,g,\Omega,J)$ denote the metric $g$, symplectic form $\Omega$, and complex structure $J$ of a Kähler manifold $M$, ...